Granular flows are ubiquitous in natural phenomena, such as landslides, debris flows, and rock falls [1]}, [2]}, [3]}, and they can exhibit different flow behaviors akin to solids, fluids, or gases [4]}, [5]}, [6]}, [7]}. Complex environmental conditions, and highly dissipative interactions make it difficult to obtain a unified constitutive law for their flow characteristics.
On one hand, following pioneering works [8]}, [9]}, [10]} on dry granular flows in steady state conditions, it has been determined that the apparent friction coefficient \(\mu \)  and the solid fraction \(\phi \)  can be considered as a function of the inertial number \(I\) . The inertial number \(I=\dot{\gamma }d/\sqrt{P/\rho _{s}}\)  is defined as a ratio of the microscopic time scale (\(\sqrt{d^{2}\rho _{s}/P}\) ) to the macroscopic deformation time scale (\(1/\dot{\gamma }\) ) [7]}, [9]}, where \(\dot{\gamma }\)  is the shear rate, \(d\)  is the average particle diameter, \(P\)  is the pressure applied to the granular sample, and \(\rho _{s}\)  is the particle density. Lacaze et al.[13]} verified this theory through transient granular column collapse experiments, showing a successful application of the \(\mu (I)\)  theory to granular flows.
